Questions M4 (327 questions)

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OCR M4 2015 June Q4
9 marks Challenging +1.8
4
  1. Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter. A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
  2. Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.
OCR M4 2015 June Q6
22 marks Challenging +1.8
6 A pendulum consists of a uniform rod \(A B\) of length \(2 a\) and mass \(2 m\) and a particle of mass \(m\) that is attached to the end \(B\). The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\).
  1. Show that the moment of inertia of this pendulum about the axis of rotation is \(\frac { 20 } { 3 } m a ^ { 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The pendulum is initially held with \(B\) vertically above \(A\) (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is \(\theta\) radians the pendulum has angular speed \(\omega \mathrm { rads } ^ { - 1 }\) (see Fig. 2).
  2. Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
  3. Find the angular acceleration of the pendulum in terms of \(g , a\) and \(\theta\). At an instant when \(\theta = \frac { 1 } { 3 } \pi\), the force acting on the pendulum at \(A\) has magnitude \(F\).
  4. Find \(F\) in terms of \(m\) and \(g\). It is given that \(a = 0.735 \mathrm {~m}\).
  5. Show that the time taken for the pendulum to move from the position \(\theta = \frac { 1 } { 6 } \pi\) to the position \(\theta = \frac { 1 } { 3 } \pi\) is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant \(k\). Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}
OCR M4 2017 June Q1
7 marks Challenging +1.2
1 A uniform rod with centre \(C\) has mass \(2 M\) and length 4a. The rod is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through a point \(A\) on the rod, where \(A C = k a\) and \(0 < k < 2\). The rod is making small oscillations about the equilibrium position with period \(T\).
  1. Show that \(T = 2 \pi \sqrt { \frac { a } { 3 g } \left( \frac { 4 + 3 k ^ { 2 } } { k } \right) }\). (You may assume the standard formula \(T = 2 \pi \sqrt { \frac { I } { m g h } }\) for the period of small oscillations of a compound pendulum.)
  2. Hence find the value of \(k ^ { 2 }\) for which the period of oscillations is least.
OCR M4 2017 June Q2
9 marks Challenging +1.2
2 A ship \(S\) is travelling with constant speed \(5 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(325 ^ { \circ }\). A second ship \(T\) observes \(S\) when \(S\) is 9500 m from \(T\) on a bearing of \(060 ^ { \circ }\) from \(T\). Ship \(T\) sets off in pursuit, travelling with constant speed \(8.5 \mathrm {~ms} ^ { - 1 }\) in a straight line.
  1. Find the bearing of the course which \(T\) should take in order to intercept \(S\).
  2. Find the distance travelled by \(S\) from the moment that \(T\) sets off in pursuit until the point of interception.
OCR M4 2017 June Q3
17 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-2_439_444_1318_822} A uniform rod \(A B\) has mass \(m\) and length \(4 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(3 a\) above \(A\). The string is always vertical and the rod makes an angle \(\theta\) radians with the horizontal, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) (see diagram).
  1. Taking \(A\) as the reference level for gravitational potential energy, find an expression for the total potential energy \(V\) of the system, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \cos \theta ( 4 \lambda ( 1 + 2 \sin \theta ) - 1 ) .$$ Determine the positions of equilibrium and the nature of their stability in the cases
  2. \(\lambda > \frac { 1 } { 12 }\),
  3. \(\lambda < \frac { 1 } { 12 }\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-3_392_689_269_671} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln x\). The region \(R\), shaded in the diagram, is bounded by the curve, the \(x\)-axis and the line \(x = 4\). A uniform solid of revolution is formed by rotating \(R\) completely about the \(y\)-axis to form a solid of volume \(V\).
  4. Show that \(V = \frac { 1 } { 4 } \pi ( 64 \ln 2 - 15 )\).
  5. Find the exact \(y\)-coordinate of the centre of mass of the solid. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_385_741_269_646} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows part of the line \(y = \frac { a } { h } x\), where \(a\) and \(h\) are constants. The shaded region bounded by the line, the \(x\)-axis and the line \(x = h\) is rotated about the \(x\)-axis to form a uniform solid cone of base radius \(a\), height \(h\) and volume \(\frac { 1 } { 3 } \pi a ^ { 2 } h\). The mass of the cone is \(M\).
  6. Show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 3 } { 20 } M \left( a ^ { 2 } + 4 h ^ { 2 } \right)\). (You may assume the standard formula \(\frac { 1 } { 4 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about a diameter.) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_501_556_1238_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A uniform solid cone has mass 3 kg , base radius 0.4 m and height 1.2 m . The cone can rotate about a fixed vertical axis passing through its centre of mass with the axis of the cone moving in a horizontal plane. The cone is rotating about this vertical axis at an angular speed of \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A stationary particle of mass \(m \mathrm {~kg}\) becomes attached to the vertex of the cone (see Fig. 2). The particle being attached to the cone causes the angular speed to change instantaneously from \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(7.8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  7. Find the value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-5_534_501_255_767} A triangular frame \(A B C\) consists of three uniform rods \(A B , B C\) and \(C A\), rigidly joined at \(A , B\) and \(C\). Each rod has mass \(m\) and length \(2 a\). The frame is free to rotate in a vertical plane about a fixed horizontal axis passing through \(A\). The frame is initially held such that the axis of symmetry through \(A\) is vertical and \(B C\) is below the level of \(A\). The frame starts to rotate with an initial angular speed of \(\omega\) and at time \(t\) the angle between the axis of symmetry through \(A\) and the vertical is \(\theta\) (see diagram).
  8. Show that the moment of inertia of the frame about the axis through \(A\) is \(6 m a ^ { 2 }\).
  9. Show that the angular speed \(\dot { \theta }\) of the frame when it has turned through an angle \(\theta\) satisfies $$a \dot { \theta } ^ { 2 } = a \omega ^ { 2 } - k g \sqrt { 3 } ( 1 - \cos \theta ) ,$$ stating the exact value of the constant \(k\).
    Hence find, in terms of \(a\) and \(g\), the set of values of \(\omega ^ { 2 }\) for which the frame makes complete revolutions. At an instant when \(\theta = \frac { 1 } { 6 } \pi\), the force acting on the frame at \(A\) has magnitude \(F\).
  10. Given that \(\omega ^ { 2 } = \frac { 2 g } { a \sqrt { 3 } }\), find \(F\) in terms of \(m\) and \(g\). \section*{END OF QUESTION PAPER}
OCR MEI M4 Q2
12 marks Challenging +1.8
2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5bed3ad4-0e20-4458-a37f-655faf84c31a-02_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
  2. Hence find the positions of equilibrium and investigate their stability.
OCR MEI M4 2006 June Q1
12 marks Challenging +1.8
1 A spherical raindrop falls through a stationary cloud. Water condenses on the raindrop and it gains mass at a rate proportional to its surface area. At time \(t\) the radius of the raindrop is \(r\). Initially the raindrop is at rest and \(r = r _ { 0 }\). The density of the water is \(\rho\).
  1. Show that \(\frac { \mathrm { d } r } { \mathrm {~d} t } = k\), where \(k\) is a constant. Hence find the mass of the raindrop in terms of \(r _ { 0 } , \rho , k\) and \(t\).
  2. Assuming that air resistance is negligible, find the velocity of the raindrop in terms of \(r _ { 0 } , k\) and \(t\).
OCR MEI M4 2006 June Q2
12 marks Challenging +1.2
2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-2_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
  2. Hence find the positions of equilibrium and investigate their stability.
OCR MEI M4 2006 June Q3
24 marks Challenging +1.8
3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, \(P\) watts, modelled by $$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$ where \(m \mathrm {~kg}\) is the mass of the aeroplane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity at time \(t\) seconds. The displacement of the aeroplane from its starting point is \(x \mathrm {~m}\). To take off successfully the aeroplane must reach a speed of \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it has travelled 900 m .
  1. Formulate and solve a differential equation for \(v\) in terms of \(x\). Hence show that the aeroplane takes off successfully.
  2. Formulate a differential equation for \(v\) in terms of \(t\). Solve the differential equation to show that \(v = 100 \tan ( 0.04 t )\). What feature of this result casts doubt on the validity of the model?
  3. In fact the model is only valid for \(0 \leqslant t \leqslant 11\), after which the power remains constant at the value attained at \(t = 11\). Will the aeroplane take off successfully?
OCR MEI M4 2006 June Q4
24 marks Challenging +1.8
4 A flagpole AB of length \(2 a\) is modelled as a thin rigid rod of variable mass per unit length given by $$\rho = \frac { M } { 8 a ^ { 2 } } ( 5 a - x ) ,$$ where \(x\) is the distance from A and \(M\) is the mass of the flagpole.
  1. Show that the moment of inertia of the flagpole about an axis through A and perpendicular to the flagpole is \(\frac { 7 } { 6 } M a ^ { 2 }\). Show also that the centre of mass of the flagpole is at a distance \(\frac { 11 } { 12 } a\) from A . The flagpole is hinged to a wall at A and can rotate freely in a vertical plane. A light inextensible rope of length \(2 \sqrt { 2 } a\) is attached to the end B and the other end is attached to a point on the wall a distance \(2 a\) vertically above A, as shown in Fig. 4. The flagpole is initially at rest when lying vertically against the wall, and then is displaced slightly so that it falls to a horizontal position, at which point the rope becomes taut and the flagpole comes to rest. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-4_403_365_1174_849} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  2. Find an expression for the angular velocity of the flagpole when it has turned through an angle \(\theta\).
  3. Show that the vertical component of the impulse in the rope when it becomes taut is \(\frac { 1 } { 12 } M \sqrt { 77 a g }\). Hence write down the horizontal component.
  4. Find the horizontal and vertical components of the impulse that the hinge exerts on the flagpole when the rope becomes taut. Hence find the angle that this impulse makes with the horizontal.
OCR MEI M4 2007 June Q1
12 marks Challenging +1.2
1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance \(a\) from the pole and has a smooth ring of mass \(m\) attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance \(y\) below the horizontal level of the peg. This situation is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-2_462_275_557_897} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The string has stiffness \(k\) and natural length equal to the distance AP .
  1. Express the extension of the string in terms of \(y\) and \(a\). Hence find the potential energy of the system relative to the level of P .
  2. Use the potential energy to find the equilibrium position of the system, and show that it is stable.
  3. Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.
OCR MEI M4 2007 June Q2
12 marks Challenging +1.2
2 A railway truck of mass \(m _ { 0 }\) travels along a horizontal track. There is no driving force and the resistances to motion are negligible. The truck is being filled with coal which falls vertically into it at a mass rate \(k\). The process starts as the truck passes a point O with speed \(u\). After time \(t\), the truck has velocity \(v\) and the displacement from O is \(x\).
  1. Show that \(v = \frac { m _ { 0 } u } { m _ { 0 } + k t }\) and find \(x\) in terms of \(m _ { 0 } , u , k\) and \(t\).
  2. Find the distance that the truck has travelled when its speed has been halved.
OCR MEI M4 2007 June Q3
24 marks Challenging +1.3
3
  1. Show, by integration, that the moment of inertia of a uniform rod of mass \(m\) and length \(2 a\) about an axis through its centre and perpendicular to the rod is \(\frac { 1 } { 3 } m a ^ { 2 }\). A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod. The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
  3. The pendulum is held with AB at an angle \(\alpha\) to the downward vertical and released from rest. At time \(t , \mathrm { AB }\) is at an angle \(\theta\) to the vertical. Find an expression for \(\dot { \theta } ^ { 2 }\) in terms of \(\theta\) and \(\alpha\).
  4. Hence, or otherwise, show that, provided that \(\alpha\) is small, the pendulum performs simple harmonic motion. Calculate the period.
OCR MEI M4 2007 June Q4
24 marks Challenging +1.8
4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) under the action of a force \(F \mathrm {~N}\), where \(F = 2 - 8 v ^ { 2 }\). The displacement of the particle from O at time \(t\) seconds is \(x \mathrm {~m}\).
  1. Formulate and solve a differential equation to show that \(v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)\).
  2. Hence express \(F\) in terms of \(x\) and find, by integration, the work done in the first 2 m of the motion.
  3. Formulate and solve a differential equation to show that \(v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)\).
  4. Calculate \(v\) when \(t = 1\) and when \(t = 2\), giving your answers to four significant figures. Hence find the impulse of the force \(F\) over the interval \(1 \leqslant t \leqslant 2\).
OCR MEI M4 2008 June Q1
12 marks Challenging +1.2
1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).
  1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
  2. Hence find an expression for \(v\) in terms of \(t\).
  3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).
OCR MEI M4 2008 June Q2
12 marks Challenging +1.2
2 A car of mass \(m \mathrm {~kg}\) starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power \(P\) watts, given by \(P = m \left( k ^ { 2 } - v ^ { 2 } \right)\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the car and \(k\) is a positive constant. The displacement from O in the direction of motion is \(x \mathrm {~m}\).
  1. Show that \(\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\), and hence find \(x\) in terms of \(v\) and \(k\).
  2. How far does the car travel before reaching \(90 \%\) of its terminal velocity?
OCR MEI M4 2008 June Q3
24 marks Challenging +1.2
3 A circular disc of radius \(a \mathrm {~m}\) has mass per unit area \(\rho \mathrm { kg } \mathrm { m } ^ { - 2 }\) given by \(\rho = k ( a + r )\), where \(r \mathrm {~m}\) is the distance from the centre and \(k\) is a positive constant. The disc can rotate freely about an axis perpendicular to it and through its centre.
  1. Show that the mass, \(M \mathrm {~kg}\), of the disc is given by \(M = \frac { 5 } { 3 } k \pi a ^ { 3 }\), and show that the moment of inertia, \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), about this axis is given by \(I = \frac { 27 } { 50 } M a ^ { 2 }\). For the rest of this question, take \(M = 64\) and \(a = 0.625\).
    The disc is at rest when it is given a tangential impulsive blow of 50 N s at a point on its circumference.
  2. Find the angular speed of the disc. The disc is then accelerated by a constant couple reaching an angular speed of \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 20 seconds.
  3. Calculate the magnitude of this couple. When the angular speed is \(30 \mathrm { rads } ^ { - 1 }\), the couple is removed and brakes are applied to bring the disc to rest. The effect of the brakes is modelled by a resistive couple of \(3 \dot { \theta } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the disc in \(\mathrm { rad } \mathrm { s } ^ { - 1 }\).
  4. Formulate a differential equation for \(\dot { \theta }\) and hence find \(\dot { \theta }\) in terms of \(t\), the time in seconds from when the brakes are first applied.
  5. By reference to your expression for \(\dot { \theta }\), give a brief criticism of this model for the effect of the brakes.
OCR MEI M4 2008 June Q4
24 marks Challenging +1.2
4 A uniform smooth pulley can rotate freely about its axis, which is fixed and horizontal. A light elastic string AB is attached to the pulley at the end B . The end A is attached to a fixed point such that the string is vertical and is initially at its natural length with B at the same horizontal level as the axis. In this position a particle P is attached to the highest point of the pulley. This initial position is shown in Fig. 4.1. The radius of the pulley is \(a\), the mass of P is \(m\) and the stiffness of the string AB is \(\frac { m g } { 10 a }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_451_517_607_466} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_456_451_607_1226} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure}
  1. Fig. 4.2 shows the system with the pulley rotated through an angle \(\theta\) and the string stretched. Write down the extension of the string and hence find the potential energy, \(V\), of the system in this position. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \left( \frac { 1 } { 10 } \theta - \sin \theta \right)\).
  2. Hence deduce that the system has a position of unstable equilibrium at \(\theta = 0\).
  3. Explain how your expression for \(V\) relies on smooth contact between the string and the pulley. Fig. 4.3 shows the graph of the function \(\mathrm { f } ( \theta ) = \frac { 1 } { 10 } \theta - \sin \theta\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_538_1342_1706_404} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
    \end{figure}
  4. Use the graph to give rough estimates of three further values of \(\theta\) (other than \(\theta = 0\) ) which give positions of equilibrium. In each case, state with reasons whether the equilibrium is stable or unstable.
  5. Show on a sketch the physical situation corresponding to the least value of \(\theta\) you identified in part (iv). On your sketch, mark clearly the positions of P and B .
  6. The equation \(\mathrm { f } ( \theta ) = 0\) has another root at \(\theta \approx - 2.9\). Explain, with justification, whether this necessarily gives a position of equilibrium.
OCR MEI M4 2009 June Q1
12 marks Challenging +1.8
1 A raindrop increases in mass as it falls vertically from rest through a stationary cloud. At time \(t \mathrm {~s}\) the velocity of the raindrop is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its mass is \(m \mathrm {~kg}\). The rate at which the mass increases is modelled as \(\frac { m g } { 2 ( v + 1 ) } \mathrm { kg } \mathrm { s } ^ { - 1 }\). Resistances to motion are neglected.
  1. Write down the equation of motion of the raindrop. Hence show that $$\left( 1 - \frac { 1 } { v + 2 } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 2 } g .$$
  2. Solve this differential equation to find an expression for \(t\) in terms of \(v\). Calculate the time it takes for the velocity of the raindrop to reach \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Describe, with reasons, what happens to the acceleration of the raindrop for large values of \(t\).
OCR MEI M4 2009 June Q2
12 marks Challenging +1.2
2 A uniform rigid rod AB of mass \(m\) and length \(4 a\) is freely hinged at the end A to a horizontal rail. The end B is attached to a light elastic string BC of modulus \(\frac { 1 } { 2 } m g\) and natural length \(a\). The end C of the string is attached to a ring which is small, light and smooth. The ring can slide along the rail and is always vertically above B . The angle that AB makes below the rail is \(\theta\). The system is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9763e6c4-e372-46ef-a666-3ccb185aa5d2-2_277_707_1398_717} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Find the potential energy, \(V\), of the system when the string is stretched and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$
  2. Hence find any positions of equilibrium of the system and investigate their stability.
OCR MEI M4 2009 June Q3
24 marks Challenging +1.2
3 A uniform circular disc has mass \(M\) and radius \(a\). The centre of the disc is at point C .
  1. Show by integration that the moment of inertia of the disc about an axis through C and perpendicular to the disc is \(\frac { 1 } { 2 } M a ^ { 2 }\). The point A on the disc is at a distance \(\frac { 1 } { 10 } a\) from its centre.
  2. Show that the moment of inertia of the disc about an axis through A and perpendicular to the disc is \(0.51 M a ^ { 2 }\). The disc can rotate freely in a vertical plane about an axis through A that is horizontal and perpendicular to the disc. The disc is held slightly displaced from its stable equilibrium position and is released from rest. In the motion that follows, the angle that AC makes with the downward vertical is \(\theta\).
  3. Write down the equation of motion for the disc. Assuming \(\theta\) remains sufficiently small throughout the motion, show that the disc performs approximate simple harmonic motion and determine the period of the motion. A particle of mass \(m\) is attached at a point P on the circumference of the disc, so that the centre of mass of the system is now at A .
  4. Sketch the position of P in relation to A and C . Find \(m\) in terms of \(M\) and show that the moment of inertia of the system about the axis through A and perpendicular to the disc is \(0.6 M a ^ { 2 }\). The system now rotates at a constant angular speed \(\omega\) about the axis through A .
  5. Find the kinetic energy of the system. Hence find the magnitude of the constant resistive couple needed to bring the system to rest in \(n\) revolutions.
OCR MEI M4 2009 June Q4
24 marks Challenging +1.2
4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion \(R \mathrm {~N}\). At time \(t \mathrm {~s}\) the velocity of the parachutist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance she has fallen is \(x \mathrm {~m}\). While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as \(R = k v ^ { 2 }\), where \(k\) is a constant. The terminal velocity of the parachutist in free-fall is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = \frac { g } { 40 }\).
  2. Show that \(v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)\). When she has fallen 1800 m , she opens her parachute.
  3. Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist. As the parachute opens, the resistance instantly changes and is now modelled as \(R = 90 v\).
  4. Calculate her velocity just before opening the parachute, correct to four decimal places.
  5. Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR MEI M4 2010 June Q1
12 marks Challenging +1.2
1 At time \(t\) a rocket has mass \(m\) and is moving vertically upwards with velocity \(v\). The propulsion system ejects matter at a constant speed \(u\) relative to the rocket. The only additional force acting on the rocket is its weight.
  1. Derive the differential equation \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + u \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The rocket has initial mass \(m _ { 0 }\) of which \(75 \%\) is fuel. It is launched from rest. Matter is ejected at a constant mass rate \(k\).
  2. Assuming that the acceleration due to gravity is constant, find the speed of the rocket at the instant when all the fuel is burnt.
OCR MEI M4 2010 June Q2
12 marks Standard +0.8
2 A particle of mass \(m \mathrm {~kg}\) moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The total resistance force on the particle is of magnitude \(m k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\) where \(k\) is a positive constant. There are no other horizontal forces present. Initially \(v = 25\) and the particle is at a point O .
  1. Show that \(v = 4 \left( k t + \frac { 2 } { 5 } \right) ^ { - 2 }\).
  2. Find the displacement from O of the particle at time \(t\).
  3. Describe the motion of the particle as \(t\) increases. Section B (48 marks)
OCR MEI M4 2010 June Q3
24 marks Challenging +1.2
3 A uniform rod AB of mass \(m\) and length \(4 a\) is hinged at a fixed point C , where \(\mathrm { AC } = a\), and can rotate freely in a vertical plane. A light elastic string of natural length \(2 a\) and modulus \(\lambda\) is attached at one end to B and at the other end to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(2 a\) above C . The string is always vertical. This system is shown in Fig. 3 with the rod inclined at \(\theta\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-2_387_613_1763_767} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find an expression for \(V\), the potential energy of the system relative to C , and show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( \frac { 9 } { 2 } \lambda \sin \theta - m g \right)\).
  2. Determine the positions of equilibrium and the nature of their stability in the cases
    (A) \(\lambda > \frac { 2 } { 9 } m g\),
    (B) \(\lambda < \frac { 2 } { 9 } m g\),
    (C) \(\lambda = \frac { 2 } { 9 } m g\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_522_755_342_696} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure}
  3. Show, by integration, that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M a ^ { 2 }\). [You may assume the standard formula for the moment of inertia of a uniform circular disc about its axis of symmetry and the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] A frustum is made by taking a uniform cone of base radius 0.1 m and height 0.2 m and removing a cone of height 0.1 m and base radius 0.05 m as shown in Fig. 4.2. The mass of the frustum is 2.8 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_391_517_1352_813} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure} The frustum can rotate freely about its axis of symmetry which is fixed and vertical.
  4. Show that the moment of inertia of the frustum about its axis of symmetry is \(0.0093 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The frustum is accelerated from rest for \(t\) seconds by a couple of magnitude 0.05 N m about its axis of symmetry, until it is rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  5. Calculate \(t\).
  6. Find the position of G , the centre of mass of the frustum. The frustum (rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) ) then receives an impulse tangential to the circumference of the larger circular face. This reduces its angular speed to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  7. To reduce its angular speed further, a parallel impulse of the same magnitude is now applied tangentially in the horizontal plane through G at the curved surface of the frustum. Calculate the resulting angular speed.